Non-existence of natural states for Abelian Chern–Simons theory

We give an elementary proof that Abelian Chern–Simons theory, described as a functor from oriented surfaces to C∗-algebras, does not admit a natural state. Non-existence of natural states is thus not only a phenomenon of quantum field theories on Lorentzian manifolds, but also of topological quantum field theories formulated in the algebraic approach.